Non-stationary response of a beam to a moving random force

The non-stationary random vibration of a beam is investigated. The beam is subjected to a random force with constant mean value which is moving with constant speed along the beam. The statistical characteristics of the first and second order for the deflection and bending moment of the beam are computed by using the correlation method. The numerical results of the coefficient of variation of the deflection at beam span mid-point are given for five basic types of convariances of the force (white noise, constant, exponential cosine, exponential, and cosine wave). The effect of the speed of the movement of the force along the beam as well as the effect of the beam damping is investigated in detail. It is concluded that the resulting beam vibration turns out to be a non-stationary process even though the motion considered is that of a stationary random force.     

Frequency domain response of a parametrically excited riser under random wave forces

Floating Production, Drilling, Storage and Offloading units represent a new technology with a promising future in the offshore oil industry. An important role is played by risers, which are installed between the subsea wellhead and the Tension Leg Deck located in the middle of the moon-pool in the hull. The inevitable heave motion of the floating hull causes a time-varying axial tension in the riser. This time dependent tension may have an undesirable influence on the lateral deflection response of the riser, with random wave forces in the frequency domain. To investigate this effect, a riser is modeled as a Bernoulli–Euler beam. The axial tension is expressed as a static part, along with a harmonic dynamic part. By linearizing the wave drag force, the riser's lateral deflection is obtained through a partial differential equation containing a time-dependent coefficient. Applying the Galerkin method, the equation is reduced to an ordinary differential equation that can be solved using the pseudo-excitation method in the frequency domain. Moreover, the Floquet–Liapunov theorem is used to estimate the stability of the vibration system in the space of parametric excitation. Finally, stability charts are obtained for some numerical examples, the correctness of the proposed method is verified by comparing with Monte-Carlo simulation and the influence of the parametric excitation on the frequency domain responses of the riser is discussed.     

VIBRATION SUPPRESSION OF A BEAM STRUCTURE BY INTERMEDIATE MASSES AND SPRINGS

A method based on the dynamic Green function has been proposed to determine the optimum values of masses and/or springs and their locations on a beam structure in order to confine the vibration at an arbitrary location. In the analysis, the beam is driven by a harmonic external excitation. The added masses on the beam and the springs attached are modelled as simple reactions that provide transverse forces to the beam. These forces act as secondary forces that reduce the response caused by the external force. Numerical simulation shows that the vibration of the beam can be confined in a certain region by the presence of masses and springs in best arrangement. This method is demonstrated for both a simply supported and a cantilever beam. An experimental set-up was designed in which a simply supported beam is excited by an electrodynamic shaker and the response of the beam is measured using an He-Ne laser system. This assures very accurate measurements and avoids any additional loading effects as in the case of accelerometers. Comparisons of the theoretical and the experimental results show good agreement.     

Forced vibration of a damped sandwich beam subjected to moving forces

The response of a sandwich beam subjected to moving forces (constant as well as pulsating) is analyzed by the use of Fourier and Laplace transforms and compared with the response of an equivalent elastic beam. The results indicate that the critical speed of force on a sandwich beam is always greater than that on an elastic beam of identical mass per unit length and flexural rigidity, and depends on its geometric and shear parameters. For subcritical speeds, the maximum deflection of a sandwich beam is shown to occur earlier than that of an equivalent elastic beam. An increase in the core shear stiffness is shown to be beneficial in reducing the dynamic magnification of the central deflection of the sandwich beam.     

Dynamic stability of spinning pretwisted beams subjected to axial random forces

This paper studies the dynamic stability of a pretwisted cantilever beam spinning along its longitudinal axis and subjected to an axial random force at the free end. The axial force is assumed as the sum of a constant force and a random process with a zero mean. Due to this axial force, the beam may experience parametric random instability. In this work, the finite element method is first applied to yield discretized system equations. The stochastic averaging method is then adopted to obtain Ito's equations for the response amplitudes of the system. Finally the mean-square stability criterion is utilized to determine the stability condition of the system. Numerical results show that the stability boundary of the system converges as the first three modes are taken into calculation. Before the convergence is reached, the stability condition predicted is not conservative enough.     

Optimal distributed control of a continuous beam with damping

Optimal control of a damped two-span beam is studied with the objective of minimizing its deflection and velocity in a given period of time with the minimum possible expenditure of force. The beam undergoes transient vibrations and is subject to given displacement and velocity initial conditions. The control is exercised by means of a transverse distributed force. The multiple objectives of the problem lead to a vector performance criterion which is reduced to a scalar one by using the concept of Pareto optimality. Necessary and sufficient conditions of optimality are expressed in the form of a maximum principle which leads to an explicit expression for the control force. The behaviour of the controlled beam is numerically studied which indicates that optimally controlled distributed forces are effective in damping out the dynamic response. Relations between various objectives are studied by means of optimal trade-off curves showing the best performance of the controlled beam.     

Vibration control of beams on elastic foundation under a moving vehicle and random lateral excitations

The formulation of three-dimensional dynamic behavior of a Beam On Elastic Foundation (BOEF) under moving loads and a moving mass is considered. The weight of the vehicle is modeled as a moving point load, however the effect of the lateral excitation is considered by modeling: (case 1) a lateral moving load with random intensity for wind excitation and (case 2) a moving mass just in lateral direction of the beam for earthquake excitation. A Dirac-delta function is used to describe the position of the moving load and the moving mass along the beam. The beam foundations are considered as elastic Winkler-type in two perpendicular transverse directions. This model is proposed to investigate the bending response of the rails under the effect of traveling vehicle weight while a random excitation such as earthquake or wind takes place. The results showed the importance of considering the effect of earthquake/wind actions as in bending stress of the beam on elastic foundations. The effect of different regions (different support stiffness) and different velocities of the vehicle on the response of the beam are investigated in mentioned directions. At the end, a linear optimal control algorithm with displacement–velocity feedback is proposed as a solution to suppress the response of BOEFs. By the method of modal analyses and taking into account enough number of vibration modes, state-space equation is obtained, then sufficient number of actuators was chosen for each direction. Stochastic analyses were performed in lateral direction in order to illustrate a comprehensive view for the response of the beam under the random moving load in both controlled and uncontrolled systems. Furthermore, the efficiency of control algorithm on critical velocities is verified by parametric analyses in the vertical direction with the constant moving load for different regions.     

Dynamic stability of rotor-bearing systems subjected to random axial forces

This paper investigates the lateral vibration of a spinning disk-shaft system supported by a pair of ball bearings and subjected to a pair of random axial forces at both ends. The axial forces are assumed as the sum of a static force and a random process with a zero mean. Due to the random axial forces, the rotor-bearing system may experience parametric random instability under certain situations. In this work, the finite element method is applied to yield a set of discretized system equations first. The set of discretized system equations is partially uncoupled by the modal analysis procedure suitable for gyroscopic systems. The stochastic averaging method is then adopted to obtain Ito's equations for the response amplitudes of the system. Finally the first- and second-moment stability criteria are utilized to determine the stability boundaries of the system. Numerical results show that the rotor-bearing system is always stable in the sense of the first-moment stability, and the effects of the average axial compressive force and the disk mass, which will lower all frequencies of the system, tend to destabilize the second-moment stability of the system.     

The vibration characteristics of a beam with an axial force

Equations of motion are found for a non-uniform damped Timoshenko beam with a distributed axial force. Principal modes may be extracted by numerical means when the boundary conditions are specified, and the appropriate orthogonality conditions are given. The theory of linear forced vibration can thus be derived. It is an implicit requirement that all axial forces are conservative. That is to say, tangential, follower and partial follower axial forces (whether applied at an extremity or distributed along the beam) are excluded.     

The energy injection into a fluid by stochastic volume forces and random stirring forces

The wavenumber spectrum of the stationary energy injection rate into an incompressible fluid described by the Navier-Stokes equations is evaluated for some simple realizations of stochastic volume as well as stirring forces. A general relation between energy injection, fluid's response, and force correlations is derived which was previously shown to be particularly simple for Gaussian distributed forces with white noise frequency spectrum. For two kinds of such model volume forces the energy injection rates are calculated: Fluid volume elements of variable size around randomly chosen positions are forced in one model centralsymmetrically in the other one anti-symmetrically under inversion with various force density profiles. The circumstances under which both models display an energy injection rate k ^–1 into a bandd k around the wavenumberk are discussed. As a simple realization of stochastic stirring forces externally moved hard spheres immersed in the fluid are considered. The equation of motion and energy balance for the velocity field of the combined system is discussed. The spectral distribution of energy injection by stirring is shown to be that of a volume force model.     

An approximate method of predicting the response of periodically supported beams subjected to random convected loading

The space-averaged response of an infinite, elastically supported, periodic beam subjected to convected random loading has been studied by using an approximate “assumed mode” method. The complex wave motion in the beam is represented by any number of suitably chosen complex modes. With a good, yet simple, choice of mode which satisfies certain boundary conditions on one periodic beam element, a “single mode approximation” can yield very accurate values of the average response. This has been verified for a wide range of the support stiffnesses and loading convection velocities. Consideration has also been given to the ratio of the maximum response in the beam to the space-averaged response. The method has been applied only to uniform beams in this paper, but it should be readily applicable to periodic systems consisting of non-uniform beam elements.     

激光对含偏心核球形粒子的辐射俘获力

利用偏心球形粒子对任意角度入射有形波束散射的理论,从广义米理论出发,根据电磁场的动量守恒及麦克斯韦张量,推导了任意入射波束对偏心球形粒子辐射俘获力的级数表达式,并以高斯波束为例,就离轴入射有吸收偏心球形粒子时的辐射俘获力进行了数值模拟,讨论了束腰半径、吸收系数、内核的相对大小及位置对俘获情况的影响. 关键词： 广义米理论 偏心球 辐射俘获力 光镊     

Interaction response of maglev masses moving on a suspended beam shaken by horizontal ground motion

As a maglev transport route has to cross a region with occasional earthquakes, the train/guideway interaction is an issue of great concern in dominating safety of the maglev system. This paper intends to present a computational framework of interaction analysis for a maglev train traveling over a suspension bridge shaken by horizontal earthquakes. The suspended guideway girder is modeled as a single-span suspended beam and the maglev train traveling over it as a series of maglev masses. Due to motion-dependent nature of magnetic forces in a maglev suspension system, appropriate adjustments of the magnetic forces between magnets and guide-rail require the air gaps be continuously monitored. Thus an on-board hybrid LQR+PID controller with constraint rule base is designed to control the dynamic response of a running maglev mass. Then the governing equations of motion for the suspended beam associated with all the controlled maglev masses are transformed into a set of generalized equations by Galerkin's method, and solved using an incremental-iterative procedure. Numerical investigations demonstrate that when a controlled maglev train travels over a suspended guideway shaken by horizontal earthquakes, the proposed hybrid controller has the ability to adjust the levitation gaps in a prescribed stable region for safety reasons and to reduce the vehicle's acceleration response for ride quality.     

高斯波束对双层粒子的辐射俘获力

从广义米理论出发，将入射高斯波束用矢量球谐函数展开.根据对电磁场动量的讨论，得出了高斯波束对多层球形粒子的辐射俘获力的表示式，并就单高斯波束对在轴双层有吸收粒子受到的辐射俘获力进行了数值模拟，讨论了束腰半径、吸收系数、内外层相对厚度对俘获情况的影响. 关键词： 辐射俘获力 多层球形粒子 光镊 高斯波束     

Characteristics of a high frequency ion source used in a low-energy accelerator

This work presents the characteristics of a high frequency ion source operating on a low energy, 150 keV accelerator. The latter is to be used as a neutron generator and its design is based on a theoretical analysis which shows that if the axial potential in an electrostatic electrode system is made to increase with four thirds the power of axial distance, inward electric forces will compensate space charge forces tending to blow up the beam. This results in a simplified acceleration tube much shorter and of higher gradient than the conventional acceleration columns. The ion source itself is an ordinary type using axial extraction of the beam, and its main properties investigated are the beam current and beam quality (or emittance). Dependence of the two on different parameters is investigated in a series of tests.     

Role of interparticle forces in the formation of random loose packing

We present a physical and numerical study of the settling of uniform spheres in liquids and show that interparticle forces play a critical role in forming the so-called random loose packing (RLP). Different packing conditions give different interparticle forces and, hence, different RLP. Two types of interparticle forces are identified: process dependent and process independent. The van der Waals force, as the major cohesive force in the present study, plays a critical role in effecting the process-dependent forces such as drag and lift forces. An equation is formulated to describe the relationship between the macroscopic packing fraction and microscopic interparticle forces in a packing. We argue there is no lowest packing fraction for a mechanically stable RLP; hence, the packing fractions of RLP can range from 0 to 0.64 depending on the cohesive and frictional conditions between particles.     

Simulation of euclidean quantum field theories by a random walk process

A new Monte Carlo method for euclidean lattice field theory is introduced by writing the Boltzmann distribution e^?s as a solution of a diffusion type equation and constructing the associated random walk process. It is practically tested for a quantum mechanical model and a non-compact version of lattice QCD. It is explained where the main interest in this algorithm lies: the diffusion process coming from an action that can be generalized to include non-conservative forces. This possibility is exploited in our QCD version to implement gauge fixing without Faddeev-Popov ghosts.     

Elastic response of a free half-space to random force excitations applied on the boundary

A response of an elastic half-space to random forces applied normally to the free boundary is studied. This paper is the second part of the study we presented in [I.A. Shalimova, K.K. Sabelfeld, The response of an elastic three-dimensional half-space to random correlated displacement perturbations on the boundary, Physica A 389 (21) (2010) 4436–4449] where the case of random displacements on the boundary was considered. We analyze the white noise excitations in detail, and derive explicitly the mean of the elastic energy, the strain and displacement correlation tensors. Simulation algorithms are constructed both for displacement and strain random fields, which enables us to calculate any desired statistical characteristics.     

Dynamics of a laminated composite beam with delamination and inclusions

In this work numerical and experimental study of the dynamic behaviour of a composite laminated beam having delamination is presented. The model of delamination takes into account a contact interaction between sublaminates including normal forces, shear forces and additional damping. In order to verify the model special samples of multilayered beams have been manufactured. Small parts of adjacent layers have been cut and replaced by inclusions from different materials modelling delamination. The mechanical properties of the inclusions have been considered during the numerical calculations. The beams were subjected to a short pulse loading and then their response was registered. The results from the numerical simulation were in a good agreement with the experimental results. The significance of the additional damping due to delamination on the response of the beam was confirmed numerically and experimentally.